By law, a box of cereal labeled as containing 36 ounces must contain at least 36 ounces of cereal. The machine filling the boxes produces a distribution of fill weights with a mean equal to the setting on the machine and with a standard deviation equal to 0.02 ounce. To ensure that most of the boxes contain at least 36 ounces, the machine is set so that the mean fill per box is 36.06 ounces.
Assuming nothing is known about the shape of the distribution, what can be said about the proportion of cereal boxes that contain less than 36 ounces.
A) The proportion is at most 11%.
B) The proportion is at least 89%.
C) The proportion is less than 2.5%.
D) The proportion is at most 5.5%.

Question

Quizplus · Accepted Answer

Using Chebyshev's inequality, which applies regardless of the distribution's shape, at least 75% of the data lies within 2 standard deviations from the mean. Since the mean is 36.06 and the standard deviation is 0.02, 36 ounces is 3 standard deviations below the mean. Therefore, at most 1/9 (about 11%) of the boxes can contain less than 36 ounces.

By Law, a Box of Cereal Labeled as Containing 36